Blackbox Reductions from Mechanisms to Algorithms

1. Blackbox Reductions fromMechanisms to Algorithms Nicole Immorlica, Northwestern U. & MSR

2. Algorithm Design Input v v1 • v2 MACHINE SCHEDULING Feasibility constraintson outcome space SET COVER ASSIGNMENT Output x • v3 • v4 • v5 GOAL: maximize (or minimize) some function f(x,v)

3. Mechanism Design bi chosen to maximize utility = vixi(b)-pi(b) Input v Input b v1 b1 • v2 • b2 MACHINE SCHEDULING Allocation x Feasibility constraintson outcome space ASSIGNMENT SET COVER • v3 • b3 Payment p • v4 • b4 • v5 • b5 GOAL: maximize (or minimize) some function f(x,v)

4. Algorithmic Mechanism Design: behind every great mechanism is a great algorithm computation incentives

5. HOLY GRAIL: general technique to convert algorithms into mechanisms

6. Black-Box Transformations Algorithm Transformation Allocation x Input v Input b Payment p GOAL: for every algorithm, transformation preserves quality of solutionin equilibrium. and is incentive compatible.

7. Black-Box Transformations Algorithm Transformation Input v Allocation x … and is incentive compatible (IC), i.e., monotone: ex-post IC (truthful in expectation):allocation to agent iis increasing in i’s bid for all bid profiles of others Bayesian IC: allocation to agent i is increasing in i’s bid in expectation w.r.t. prior of over bid profiles of others

8. Optimal Algorithm VCG Transformation Input v Allocation x EXAMPLE: Vickrey-Clark-Groves auction transforms any optimal algorithm into an optimal ex-post IC mechanism for any monotone objective function.

9. Single Item Auction SelectionAlgorithm Find agentw/max value VCG Transformation Input v Allocation x one item, agent i has value vi for item (single-parameter)

10. Combinatorial Auction agents items i1 $10 $12 • i2 • i2 • i3 many items, agent i has value vij for subset Sj (multi-parameter)

11. Combinatorial Auction ??? Find max valuenon-overlappingcollection of sets VCG Transformation Input v Allocation x many items, agent i has value vij for subset Sj (multi-parameter)

12. HOLY GRAIL: general technique to convert algorithms into mechanisms ^ approximation

13. Bayesian IC transformations for welfare (single- or multi-parameter!) polynomial time? truthful transformations for welfare Bayesian IC transformations for non-linear objectives (see Shuchi’s talk) = max ∑ixivis.t. feasibility constraints on alloc. x

14. BIC Transformation Positive Result: Transform approximation algorithms into Bayesian IC mechs with small loss in social welfare. Single-parameter: (single private value for allocation) Monotonization.For dist. F and algorithm A, there is a Bayesian IC transformation TA,F satisfying E[TA,F(v)] ≥ E[A(v)]. Blackbox computation.TA,F can be computed in polytime with queries to A. Payment computation.Payments can be computed with two queries to A.

15. Monotonization xi(vi) = E[alloc. to i| vi] Not BIC BIC vi Fact. There’re payments that make an alg. Bayesian IC if and only if for all i, expected allocation is monotone non-decreasing in value vi.

16. Monotonization Goal: construct yi from xis.t. • Monotonicity. yi(.) non-decreasing monotone • Surplus-preservation. Evi[viyi(vi)] ≥ Evi[vixi(vi)] • Distribution-preservation. (can apply construction independently to each j)

17. Monotonization Idea 1: remap values.

18. Monotonization Idea 2: resample values.

19. Monotonization allocation cumulativecurve Idea 3: resample values in region wherecumulative allocation is not monotone.

20. Monotonization xi(vi) yi(vi) Construction of yi(vi) from xi(vi) preserves: • Distribution-preservation. • Monotonicity. yinon-decreasing monotone

21. Monotonization xi(vi) yi(vi) Construction of yi(vi) from xi(vi) preserves: • Surplus-preservation. Evi[vi(yi - xi)] ≥ 0 b E[v(y-x)] = ∫ v(y-x) d f(v) (integration by parts) = v(Y-X)| – ∫ v’(Y-X) d f(v) (v , X dominates Y) = 0 – (non-neg.) x (non-pos.) (2nd term non-pos.) ≥ 0 a b b a a

22. BIC Transformation for Welfare Positive Result: Transform approximation algorithms into Bayesian IC mechs with small loss in social welfare. Single-parameter: (single private value for allocation) Monotonization.For dist. F and algorithm A, there is a Bayesian IC transformation TA,F satisfying E[A(v)] ≥ E[TA,F(v)]. Blackbox computation.TA,F can be computed in polytime with queries to A. Payment computation.Payments can be computed with two queries to A.

23. Blackbox Computation

24. BIC Transformation for Welfare Positive Result: Transform approximation algorithms into Bayesian IC mechs with small loss in social welfare. Single-parameter: (single private value for allocation) Monotonization.For dist. F and algorithm A, there is a Bayesian IC transformation TA,F satisfying E[A(v)] ≥ E[TA,F(v)]. Blackbox computation.TA,F can be computed in polytime with queries to A. Payment computation.Payments can be computed with two queries to A.

25. Payment Computation v p(v) = v y(v) – ∫ y(z)dz 0 payment identity Idea: compute random variable P with E[P] = p(v)

26. Payment Computation payment identity v p(v) = v y(v) – ∫ y(z)dz 0 Y indicator random variable for whether agent wins in A (with y(v)) z drawn uniformly from [0,v] Yz indicator random variable for whether agent wins in A (with y(z)) P = v (Y – Yz) 1st call to A 2nd call to A const. # calls per agent Idea: compute random variable P with E[P] = p(v)

27. Payment Computation goal: given A, find an alg. A’ that computesallocation and payments with just 1 call to A Pick agent k uniformly at random and draw wk from Fk Calculate outcome y’ for A(wk, v-k) For each agent i ≠ k, set p’i = viy’i For agent k, set p’k = 0 if wk > vk and p’k = -(n – 1)y’k/fk(wk) otherwise Output (y’, p’) only call to A ugly formula

28. Payment Computation Thm. Algorithm A’ is Bayesian IC. Proof. Monotone. y’ linear transformation of y. y’(v) = (1 - 1/n) y(v) + 1/n E[y(w)] Payment Identity. v v p’(v) = v y’(v) – ∫ y’(z) dz p’(v) = (1 - 1/n) vy(v) – (1/n)(n - 1)∫ y(z) dz 0 0 payment for i ≠ k payment for i = k(see ugly formula)

29. Payment Computation Thm. Welfare is E[A’(v)] ≥ E[A(v) – max(v)] Proof. Each buyer has welfare ≥ (1 - 1/n) vy(v) Since y(v) is a probability, vy(v) ≤ max(v) Lose at most max(v) in total buyer welfare Expected payments are the same, so lose nothing in seller welfare Finds (alloc, payments) with 1 call to monotone alg. [Babaioff, Kleinberg, Slivkins’10]

30. Approx. Algorithm Dist. of values Transformation Allocation x Input v Payment p (drawn fromknown dist.) POSSIBILITY: can transform any approximation algorithm into a Bayesian IC mech. with small loss for f(x,v) = Σixivi. [Hartline-Lucier’10]

31. Multi-parameter Transformation Goal: construct allocation from algorithm s.t. • “Monotonicity”. • Surplus-preservation. • Distribution-preservation. By mapping types of an agent to surrogates in a way that preserves above properties.

32. Replicas and Surrogates replicas (drawn from F) surrogates (drawn from F) surrogate allocations max-weight original type t matching surrogatetype t’ v(t,x(t’)) x(t’) Set payment equal to VCG payment for type t. Set allocation equal to output on surrogate type profile.

33. Replicas and Surrogates Thm. Transformation is distribution-preserving. Thm. Transformation is Bayesian IC. Thm. Transformation doesn’t lose much welfare. Prf. Because replicas are “close” to matched surrogatesin values for outcomes. [Hartline, Kleinberg, Malekian’11] [Bei, Huang’11]

34. Strengthening the Result Solution concept: black-box transformations for social welfare that preserve approximation and are truthful in expectation? Social objective: black-box transformations that preserve approximation, are Bayesian IC, and work for other social objectives?

35. GOAL: Find a general technique to convert approximation algorithms into truthful mech. for social welfare. IMPOSSIBLE

36. Multi-parameter Transformations Thm. There’s no truthful in expectation mech. for combinatorial auctions with submodular valuations that guarantees a sub-linear approx. Note: there is a (1-1/e)-approximation alg. [Dughmi, Vondrak’11]

37. Single-parameter Transformations Truthful in Expectation. For all algorithms A, TAis truthful in expectation, i.e., expected allocation is monotone for all i. Worst-case approximation preserving. For all values vectors v and algorithms A, expected welfare of transformation is close to expected welfare of algorithm.

38. BAD NEWS: For any polytime truthful in expectation transformation, there is a welfare problem and alg. such that worst-case welfare of transformation is polynomially larger than the alg.’s.

39. Proof Outline • Define welfare instance (feasible allocations, values of agents). • Find algorithm with high welfare. • Use monotonicity to show any ex-post transformation has low worst-case welfare.

40. Intuition • v1 row ave. of x1 increasing (x1,x2) • v2 Bayesian IC column ave. of x2 increasing

41. Intuition • v1 • v2 Ex-post IC

42. Intuition Input vector • v1 (.2,.2) (.1,.3) (.5,.5) (.3,.4) (.8,.7) Query Query Query Query • v2 Transformation must fix non-monotonicitiesin every row and column.

43. Intuition Make all allocations constant on these agents. (.6,.2) (.3,.3) 𝑣1 (.2,.6) (.5,.5) 𝑣2 Idea: hide non-monotonicity on high-dim. diagonal.

44. Truthful in Expectation Thm. Any truthful-in-expectation transformation loses a polynomial factor in welfare approximation. [Chawla, Immorlica, Lucier’12]

45. Blackbox Transformations. Bayesian IC transformations for welfare (single- or multi-parameter!) ex-post IC transformations for welfare Bayesian IC transformations for non-linear objectives (see Shuchi’s talk)

46. Thank You